TY - JOUR A1 - Engbert, Ralf A1 - Mergenthaler, Konstantin A1 - Sinn, Petra A1 - Pikovskij, Arkadij T1 - An integrated model of fixational eye movements and microsaccades JF - Proceedings of the National Academy of Sciences of the United States of America N2 - When we fixate a stationary target, our eyes generate miniature (or fixational) eye movements involuntarily. These fixational eye movements are classified as slow components (physiological drift, tremor) and microsaccades, which represent rapid, small-amplitude movements. Here we propose an integrated mathematical model for the generation of slow fixational eye movements and microsaccades. The model is based on the concept of self-avoiding random walks in a potential, a process driven by a self-generated activation field. The self-avoiding walk generates persistent movements on a short timescale, whereas, on a longer timescale, the potential produces antipersistent motions that keep the eye close to an intended fixation position. We introduce microsaccades as fast movements triggered by critical activation values. As a consequence, both slow movements and microsaccades follow the same law of motion; i.e., movements are driven by the self-generated activation field. Thus, the model contributes a unified explanation of why it has been a long-standing problem to separate slow movements and microsaccades with respect to their motion-generating principles. We conclude that the concept of a self-avoiding random walk captures fundamental properties of fixational eye movements and provides a coherent theoretical framework for two physiologically distinct movement types. Y1 - 2011 U6 - https://doi.org/10.1073/pnas.1102730108 SN - 0027-8424 VL - 108 IS - 39 SP - E765 EP - E770 PB - National Acad. of Sciences CY - Washington ER - TY - JOUR A1 - Burylko, Oleksandr A1 - Pikovskij, Arkadij T1 - Desynchronization transitions in nonlinearly coupled phase oscillators JF - Physica :D, Nonlinear phenomena N2 - We consider the nonlinear extension of the Kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter. A bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed. We demonstrate that for small ensembles it is typically mediated by stable cluster states, that disappear with creation of heteroclinic cycles, while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs. KW - Coupled oscillators KW - Oscillator ensembles KW - Kuramoto model KW - Nonlinear coupling KW - Bifurcations Y1 - 2011 U6 - https://doi.org/10.1016/j.physd.2011.05.016 SN - 0167-2789 VL - 240 IS - 17 SP - 1352 EP - 1361 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael T1 - Dynamics of heterogeneous oscillator ensembles in terms of collective variables JF - Physica :D, Nonlinear phenomena N2 - We consider general heterogeneous ensembles of phase oscillators, sine coupled to arbitrary external fields. Starting with the infinitely large ensembles, we extend the Watanabe-Strogatz theory, valid for identical oscillators, to cover the case of an arbitrary parameter distribution. The obtained equations yield the description of the ensemble dynamics in terms of collective variables and constants of motion. As a particular case of the general setup we consider hierarchically organized ensembles, consisting of a finite number of subpopulations, whereas the number of elements in a subpopulation can be both finite or infinite. Next, we link the Watanabe-Strogatz and Ott-Antonsen theories and demonstrate that the latter one corresponds to a particular choice of constants of motion. The approach is applied to the standard Kuramoto-Sakaguchi model, to its extension for the case of nonlinear coupling, and to the description of two interacting subpopulations, exhibiting a chimera state. With these examples we illustrate that, although the asymptotic dynamics can be found within the framework of the Ott-Antonsen theory, the transients depend on the constants of motion. The most dramatic effect is the dependence of the basins of attraction of different synchronous regimes on the initial configuration of phases. KW - Coupled oscillators KW - Oscillator ensembles KW - Kuramoto model KW - Nonlinear coupling KW - Watanabe-Strogatz theory KW - Ott-Antonsen theory Y1 - 2011 U6 - https://doi.org/10.1016/j.physd.2011.01.002 SN - 0167-2789 VL - 240 IS - 9-10 SP - 872 EP - 881 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Lueck, S. A1 - Pikovskij, Arkadij T1 - Dynamics of multi-frequency oscillator ensembles with resonant coupling JF - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics N2 - We study dynamics of populations of resonantly coupled oscillators having different frequencies. Starting from the coupled van der Pol equations we derive the Kuramoto-type phase model for the situation, where the natural frequencies of two interacting subpopulations are in relation 2 : 1. Depending on the parameter of coupling, ensembles can demonstrate fully synchronous clusters, partial synchrony (only one subpopulation synchronizes), or asynchrony in both subpopulations. Theoretical description of the dynamics based on the Watanabe-Strogatz approach is developed. KW - Oscillator populations KW - Kuramoto model KW - Resonant interaction Y1 - 2011 U6 - https://doi.org/10.1016/j.physleta.2011.06.016 SN - 0375-9601 VL - 375 IS - 28-29 SP - 2714 EP - 2719 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Komarov, Maxim A1 - Pikovskij, Arkadij T1 - Effects of nonresonant interaction in ensembles of phase oscillators JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We consider general properties of groups of interacting oscillators, for which the natural frequencies are not in resonance. Such groups interact via nonoscillating collective variables like the amplitudes of the order parameters defined for each group. We treat the phase dynamics of the groups using the Ott-Antonsen ansatz and reduce it to a system of coupled equations for the order parameters. We describe different regimes of cosynchrony in the groups. For a large number of groups, heteroclinic cycles, corresponding to a sequential synchronous activity of groups and chaotic states where the order parameters oscillate irregularly, are possible. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.84.016210 SN - 1539-3755 VL - 84 IS - 1 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Turukina, L. V. A1 - Pikovskij, Arkadij T1 - Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators JF - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics N2 - We show that a hyperbolic chaos can be observed in resonantly coupled oscillators near a Hopf bifurcation, described by normal-form-type equations for complex amplitudes. The simplest example consists of four oscillators, comprising two alternatively activated, due to an external periodic modulation, pairs. In terms of the stroboscopic Poincare map, the phase differences change according to an expanding Bernoulli map that depends on the coupling type. Several examples of hyperbolic chaos for different types of coupling are illustrated numerically. KW - Coupled oscillators KW - Hyperbolic chaos Y1 - 2011 U6 - https://doi.org/10.1016/j.physleta.2011.02.017 SN - 0375-9601 VL - 375 IS - 11 SP - 1407 EP - 1411 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Levnajic, Zoran A1 - Pikovskij, Arkadij T1 - Network reconstruction from random phase resetting JF - Physical review letters N2 - We propose a novel method of reconstructing the topology and interaction functions for a general oscillator network. An ensemble of initial phases and the corresponding instantaneous frequencies is constructed by repeating random phase resets of the system dynamics. The desired details of network structure are then revealed by appropriately averaging over the ensemble. The method is applicable for a wide class of networks with arbitrary emergent dynamics, including full synchrony. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevLett.107.034101 SN - 0031-9007 VL - 107 IS - 3 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Straube, Arthur V. A1 - Pikovskij, Arkadij T1 - Pattern formation induced by time-dependent advection JF - Mathematical modelling of natural phenomena N2 - We study pattern-forming instabilities in reaction-advection-diffusion systems. We develop an approach based on Lyapunov-Bloch exponents to figure out the impact of a spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal with the flows periodic in space that may have arbitrary time dependence. We propose a discrete in time model, where reaction, advection, and diffusion act as successive operators, and show that a mixing advection can lead to a pattern-forming instability in a two-component system where only one of the species is advected. Physically, this can be explained as crossing a threshold of Turing instability due to effective increase of one of the diffusion constants. KW - pattern formation KW - reaction-advection-diffusion equation Y1 - 2011 U6 - https://doi.org/10.1051/mmnp/20116107 SN - 0973-5348 VL - 6 IS - 1 SP - 138 EP - 148 PB - EDP Sciences CY - Les Ulis ER - TY - GEN A1 - Straube, Arthur V. A1 - Pikovskij, Arkadij T1 - Pattern formation induced by time-dependent advection T2 - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe N2 - We study pattern-forming instabilities in reaction-advection-diffusion systems. We develop an approach based on Lyapunov-Bloch exponents to figure out the impact of a spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal with the flows periodic in space that may have arbitrary time dependence. We propose a discrete in time model, where reaction, advection, and diffusion act as successive operators, and show that a mixing advection can lead to a pattern-forming instability in a two-component system where only one of the species is advected. Physically, this can be explained as crossing a threshold of Turing instability due to effective increase of one of the diffusion constants. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 575 KW - pattern formation KW - reaction-advection-diffusion equation Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-413140 SN - 1866-8372 IS - 575 SP - 138-147 ER - TY - JOUR A1 - Zhirov, O. V. A1 - Pikovskij, Arkadij A1 - Shepelyansky, Dima L. T1 - Quantum vacuum of strongly nonlinear lattices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We study the properties of classical and quantum strongly nonlinear chains by means of extensive numerical simulations. Due to strong nonlinearity, the classical dynamics of such chains remains chaotic at arbitrarily low energies. We show that the collective excitations of classical chains are described by sound waves whose decay rate scales algebraically with the wave number with a generic exponent value. The properties of the quantum chains are studied by the quantum Monte Carlo method and it is found that the low-energy excitations are well described by effective phonon modes with the sound velocity dependent on an effective Planck constant. Our results show that at low energies the quantum effects lead to a suppression of chaos and drive the system to a quasi-integrable regime of effective phonon modes. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.83.016202 SN - 1539-3755 VL - 83 IS - 1 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Kralemann, Björn A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael T1 - Reconstructing phase dynamics of oscillator networks JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We generalize our recent approach to the reconstruction of phase dynamics of coupled oscillators from data [B. Kralemann et al., Phys. Rev. E 77, 066205 (2008)] to cover the case of small networks of coupled periodic units. Starting from a multivariate time series, we first reconstruct genuine phases and then obtain the coupling functions in terms of these phases. Partial norms of these coupling functions quantify directed coupling between oscillators. We illustrate the method by different network motifs for three coupled oscillators and for random networks of five and nine units. We also discuss nonlinear effects in coupling. Y1 - 2011 U6 - https://doi.org/10.1063/1.3597647 SN - 1054-1500 VL - 21 IS - 2 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Blaha, Karen A. A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael A1 - Clark, Matthew T. A1 - Rusin, Craig G. A1 - Hudson, John L. T1 - Reconstruction of two-dimensional phase dynamics from experiments on coupled oscillators JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - Phase models are a powerful method to quantify the coupled dynamics of nonlinear oscillators from measured data. We use two phase modeling methods to quantify the dynamics of pairs of coupled electrochemical oscillators, based on the phases of the two oscillators independently and the phase difference, respectively. We discuss the benefits of the two-dimensional approach relative to the one-dimensional approach using phase difference. We quantify the dependence of the coupling functions on the coupling magnitude and coupling time delay. We show differences in synchronization predictions of the two models using a toy model. We show that the two-dimensional approach reveals behavior not detected by the one-dimensional model in a driven experimental oscillator. This approach is broadly applicable to quantify interactions between nonlinear oscillators, especially where intrinsic oscillator sensitivity and coupling evolve with time. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.84.046201 SN - 1539-3755 VL - 84 IS - 4 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Mulansky, Mario A1 - Ahnert, Karsten A1 - Pikovskij, Arkadij T1 - Scaling of energy spreading in strongly nonlinear disordered lattices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - To characterize a destruction of Anderson localization by nonlinearity, we study the spreading behavior of initially localized states in disordered, strongly nonlinear lattices. Due to chaotic nonlinear interaction of localized linear or nonlinear modes, energy spreads nearly subdiffusively. Based on a phenomenological description by virtue of a nonlinear diffusion equation, we establish a one-parameter scaling relation between the velocity of spreading and the density, which is confirmed numerically. From this scaling it follows that for very low densities the spreading slows down compared to the pure power law. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.83.026205 SN - 1539-3755 VL - 83 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Pikovskij, Arkadij A1 - Fishman, Shmuel T1 - Scaling properties of weak chaos in nonlinear disordered lattices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We study the discrete nonlinear Schrodinger equation with a random potential in one dimension. It is characterized by the length, the strength of the random potential, and the field density that determines the effect of nonlinearity. Following the time evolution of the field and calculating the largest Lyapunov exponent, the probability of the system to be regular is established numerically and found to be a scaling function of the parameters. This property is used to calculate the asymptotic properties of the system in regimes beyond our computational power. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.83.025201 SN - 1539-3755 SN - 1550-2376 VL - 83 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Mulansky, Mario A1 - Ahnert, Karsten A1 - Pikovskij, Arkadij A1 - Shepelyansky, Dima L. T1 - Strong and weak chaos in weakly nonintegrable many-body hamiltonian systems JF - Journal of statistical physics N2 - We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology. KW - Lyapunov exponent KW - Arnold diffusion KW - Chaos spreading Y1 - 2011 U6 - https://doi.org/10.1007/s10955-011-0335-3 SN - 0022-4715 VL - 145 IS - 5 SP - 1256 EP - 1274 PB - Springer CY - New York ER -